We use the notation
\(A_H(f, x):= \frac{1}{\sqrt{H}}\sum _{x< n \le x+H} f(n).\) 
By Markov's inequality, Theorem REF  implies that there exists a set of the form
\(\begin{split}\mathcal {F}_{X, H} := \lbrace  f\in \mathcal {F}_X:|\mathbb {E}_x |A_H(f, x)|^{2k} - k!| &= o_{X\rightarrow +\infty }(1)~\text{for all $k\le \log \log \log X$} \\|\mathbb {E}_x [A_H(f, x)^k A_H(f, x)^\ell ]| &= o_{X\rightarrow +\infty }(1)~\text{for all distinct $k,\ell \le \log \log \log X$}\rbrace \end{split}\) 
with
\(\nu _X(\mathcal {F}_{X, H}) = 1-o_{X\rightarrow +\infty }(1).\) 
Since the distribution \(\mathcal {CN}(0,1)\)  is determined by its moments (see e.g. [1]}), Theorem REF  follows.
